Initial notes on open/closed addressing.

c-declarations
Joshua Potter 2024-06-12 07:35:13 -06:00
parent 7664346b62
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"Basic": [

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---
title: "2024-06-11"
---
- [x] Anki Flashcards
- [x] KoL
- [x] OGS
- [ ] Sheet Music (10 min.)
- [ ] Korean (Read 1 Story)

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---
title: "2024-06-12"
---
- [x] Anki Flashcards
- [x] KoL
- [x] OGS
- [ ] Sheet Music (10 min.)
- [ ] Korean (Read 1 Story)
* Notes on [[open-addressing|open]] and [[closed-addressing|closed]] hashing.

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---
title: "2024-06-11"
---
- [x] Anki Flashcards
- [x] KoL
- [x] OGS
- [ ] Sheet Music (10 min.)
- [ ] Korean (Read 1 Story)
* Cartesian product [[set#Cancellation Laws|cancellation laws]].
* Notes on [[relations]].

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@ -27,6 +27,85 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
<!--ID: 1716396060605-->
END%%
## Symmetric Difference
Define the **symmetric difference** of sets $A$ and $B$ as $$A \mathop{\triangle} B = (A - B) \cup (B - A)$$
%%ANKI
Basic
What two operators are used in the definition of the symmetric difference?
Back: $\cup$ and $-$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1717554445662-->
END%%
%%ANKI
Basic
How is the symmetric difference of sets $A$ and $B$ denoted?
Back: $A \mathop{\triangle} B$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1717554445665-->
END%%
%%ANKI
Basic
How is $A \mathop{\triangle} B$ defined?
Back: As $(A - B) \cup (B - A)$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1717554445670-->
END%%
## Cartesian Product
Given two sets $A$ and $B$, the **Cartesian product** $A \times B$ is defined as: $$A \times B = \{\langle x, y \rangle \mid x \in A \land y \in B\}$$
%%ANKI
Basic
How is the Cartesian product of $A$ and $B$ denoted?
Back: $A \times B$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1717679397781-->
END%%
%%ANKI
Basic
Using ordered pairs, how is $A \times B$ defined?
Back: $\{\langle x, y \rangle \mid x \in A \land y \in B\}$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1717679397797-->
END%%
%%ANKI
Basic
Who is attributed the representation of points in a plane?
Back: René Descartes.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1717679397825-->
END%%
%%ANKI
Basic
Why is the Cartesian product named the way it is?
Back: It is named after René Descartes.
Reference: “Cartesian Product,” in _Wikipedia_, April 17, 2024, [https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305).
<!--ID: 1717679397836-->
END%%
%%ANKI
Basic
Suppose $x, y \in A$. What set is $\langle x, y \rangle$ in?
Back: $\mathscr{P}\mathscr{P}A$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1717679397848-->
END%%
%%ANKI
Cloze
{$x \in A$} iff {$\{x\} \subseteq A$} iff {$\{x\} \in \mathscr{P}A$}.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1717679397860-->
END%%
## Laws
The algebra of sets obey laws reminiscent (but not exactly) of the algebra of real numbers.
@ -676,32 +755,43 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
<!--ID: 1717073537007-->
END%%
## Symmetric Difference
### Cancellation Laws
Define the **symmetric difference** of sets $A$ and $B$ as $$A \mathop{\triangle} B = (A - B) \cup (B - A)$$
Let $A$, $B$, and $C$ be sets. If $A \neq \varnothing$,
* $(A \times B = A \times C) \Rightarrow B = C$
* $(B \times A = C \times A) \Rightarrow B = C$
%%ANKI
Basic
What two operators are used in the definition of the symmetric difference?
Back: $\cup$ and $-$.
What is the left cancellation law of the Cartesian product?
Back: $(A \times B = A \times C) \Rightarrow B = C$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1717554445662-->
<!--ID: 1718107987907-->
END%%
%%ANKI
Basic
How is the symmetric difference of sets $A$ and $B$ denoted?
Back: $A \mathop{\triangle} B$
$(A \times B = A \times C) \Rightarrow B = C$ is always true if what condition is satisfied?
Back: $A \neq \varnothing$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1717554445665-->
<!--ID: 1718107987918-->
END%%
%%ANKI
Basic
How is $A \mathop{\triangle} B$ defined?
Back: As $(A - B) \cup (B - A)$.
What is the right cancellation law of the Cartesian product?
Back: $(B \times A = C \times A) \Rightarrow B = C$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1717554445670-->
<!--ID: 1718107987928-->
END%%
%%ANKI
Basic
$(B \times A = C \times A) \Rightarrow B = C$ is always true if what condition is satisfied?
Back: $A \neq \varnothing$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1718107987936-->
END%%
## Bibliography

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---
title: Closed Addressing
TARGET DECK: Obsidian::STEM
FILE TAGS: hashing::closed
tags:
- hashing
---
## Overview
In **closed addressing**, a key is always stored in the bucket it's hashed to. Collisions are dealt with using separate data structures on a per-bucket basis.
%%ANKI
Basic
What does "closed" refer to in term "closed addressing"?
Back: A key is always stored in the slot it hashes to.
Reference: “Hash Tables: Open vs Closed Addressing | Programming.Guide,” accessed June 12, 2024, [https://programming.guide/hash-tables-open-vs-closed-addressing.html](https://programming.guide/hash-tables-open-vs-closed-addressing.html).
<!--ID: 1718198717474-->
END%%
%%ANKI
Basic
What does "open" refer to in term "open hashing"?
Back: A key may resides in a data structure separate from the hash table.
Reference: “Hash Tables: Open vs Closed Addressing | Programming.Guide,” accessed June 12, 2024, [https://programming.guide/hash-tables-open-vs-closed-addressing.html](https://programming.guide/hash-tables-open-vs-closed-addressing.html).
<!--ID: 1718198717484-->
END%%
%%ANKI
Cloze
{Closed} addressing is also known as {open} hashing.
Reference: “Hash Tables: Open vs Closed Addressing | Programming.Guide,” accessed June 12, 2024, [https://programming.guide/hash-tables-open-vs-closed-addressing.html](https://programming.guide/hash-tables-open-vs-closed-addressing.html).
<!--ID: 1718198717495-->
END%%
%%ANKI
Cloze
The following is an example of {closed} addressing.
![[closed-addressing.png]]
Reference: “Hash Tables: Open vs Closed Addressing | Programming.Guide,” accessed June 12, 2024, [https://programming.guide/hash-tables-open-vs-closed-addressing.html](https://programming.guide/hash-tables-open-vs-closed-addressing.html).
<!--ID: 1718198717506-->
END%%
%%ANKI
Cloze
The following is an example of {open} hashing.
![[closed-addressing.png]]
Reference: “Hash Tables: Open vs Closed Addressing | Programming.Guide,” accessed June 12, 2024, [https://programming.guide/hash-tables-open-vs-closed-addressing.html](https://programming.guide/hash-tables-open-vs-closed-addressing.html).
<!--ID: 1718198755496-->
END%%
## Bibliography
* “Hash Tables: Open vs Closed Addressing | Programming.Guide,” accessed June 12, 2024, [https://programming.guide/hash-tables-open-vs-closed-addressing.html](https://programming.guide/hash-tables-open-vs-closed-addressing.html).
* Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

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@ -121,6 +121,23 @@ Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (
<!--ID: 1716307180987-->
END%%
%%ANKI
Basic
What distinguishes direct addressing from closed and open addressing?
Back: Direct addressing isn't concerned with conflicting keys.
Reference: “Hash Tables: Open vs Closed Addressing | Programming.Guide,” accessed June 12, 2024, [https://programming.guide/hash-tables-open-vs-closed-addressing.html](https://programming.guide/hash-tables-open-vs-closed-addressing.html).
<!--ID: 1718199205862-->
END%%
%%ANKI
Basic
Direct addressing sits between what other addressing types?
Back: Open and closed addressing.
Reference: “Hash Tables: Open vs Closed Addressing | Programming.Guide,” accessed June 12, 2024, [https://programming.guide/hash-tables-open-vs-closed-addressing.html](https://programming.guide/hash-tables-open-vs-closed-addressing.html).
<!--ID: 1718199205872-->
END%%
## Bibliography
* “Hash Tables: Open vs Closed Addressing | Programming.Guide,” accessed June 12, 2024, [https://programming.guide/hash-tables-open-vs-closed-addressing.html](https://programming.guide/hash-tables-open-vs-closed-addressing.html).
* Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

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@ -137,6 +137,48 @@ Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (
<!--ID: 1716307180980-->
END%%
An **independent uniform hash function** is the ideal theoretical abstraction. For each possible input $k$ in universe $U$, an output $h(k)$ is produced randomly and independently chosen from range $\{0, 1, \ldots, m - 1\}$. Once a value $h(k)$ is chosen, each subsequent call to $h$ with the same input $k$ yields the same output $h(k)$.
%%ANKI
Basic
What is considered the ideal (though only theoretical) hash function?
Back: The independent uniform hash function.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1718197741507-->
END%%
%%ANKI
Basic
Given independent uniform hash function $h$, what about $h$ is "independent"?
Back: Each key $k$ has output $h(k)$ determined independently from other keys.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1718197741527-->
END%%
%%ANKI
Basic
Given independent uniform hash function $h$, what about $h$ is "uniform"?
Back: Every output of $h$ is equally likely to be any of the values in its range.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1718197741537-->
END%%
%%ANKI
Basic
With respect to hashing, a random oracle refers to what kind of hash function?
Back: An independent uniform hash function.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1718197741545-->
END%%
%%ANKI
Basic
Where does "memory" come into play with independent uniform hash functions?
Back: Once $h(k)$ is determined, subsequent calls to $h$ with $k$ always yield the same value.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1718197741555-->
END%%
## Bibliography
* Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

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---
title: Open Addressing
TARGET DECK: Obsidian::STEM
FILE TAGS: hashing::open
tags:
- hashing
---
## Overview
In **open addressing**, keys always reside in the hash table. Collisions are dealt with by searching for other empty buckets within the hash table.
%%ANKI
Basic
What does "closed" refer to in term "closed hashing"?
Back: A key must reside in the hash table.
Reference: “Hash Tables: Open vs Closed Addressing | Programming.Guide,” accessed June 12, 2024, [https://programming.guide/hash-tables-open-vs-closed-addressing.html](https://programming.guide/hash-tables-open-vs-closed-addressing.html).
<!--ID: 1718198717434-->
END%%
%%ANKI
Basic
What does "open" refer to in term "open addressing"?
Back: A key is not necessarily stored in the slot it hashes to.
Reference: “Hash Tables: Open vs Closed Addressing | Programming.Guide,” accessed June 12, 2024, [https://programming.guide/hash-tables-open-vs-closed-addressing.html](https://programming.guide/hash-tables-open-vs-closed-addressing.html).
<!--ID: 1718198717447-->
END%%
%%ANKI
Cloze
{Open} addressing is also known as {closed} hashing.
Reference: “Hash Tables: Open vs Closed Addressing | Programming.Guide,” accessed June 12, 2024, [https://programming.guide/hash-tables-open-vs-closed-addressing.html](https://programming.guide/hash-tables-open-vs-closed-addressing.html).
<!--ID: 1718198717455-->
END%%
%%ANKI
Cloze
The following is an example of {closed} hashing.
![[open-addressing.png]]
Reference: “Hash Tables: Open vs Closed Addressing | Programming.Guide,” accessed June 12, 2024, [https://programming.guide/hash-tables-open-vs-closed-addressing.html](https://programming.guide/hash-tables-open-vs-closed-addressing.html).
<!--ID: 1718198717464-->
END%%
%%ANKI
Cloze
The following is an example of {open} addressing.
![[open-addressing.png]]
Reference: “Hash Tables: Open vs Closed Addressing | Programming.Guide,” accessed June 12, 2024, [https://programming.guide/hash-tables-open-vs-closed-addressing.html](https://programming.guide/hash-tables-open-vs-closed-addressing.html).
<!--ID: 1718198755486-->
END%%
## Bibliography
* “Hash Tables: Open vs Closed Addressing | Programming.Guide,” accessed June 12, 2024, [https://programming.guide/hash-tables-open-vs-closed-addressing.html](https://programming.guide/hash-tables-open-vs-closed-addressing.html).
* Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

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@ -621,8 +621,8 @@ END%%
%%ANKI
Basic
Execution of `b[i] := e` of array $b$ yields what new value of $b$?
Back: $b = (b; i{:}e)$
Execution of `b[i] := e` of array $b$ in state $s$ yields what new value of $b$?
Back: $b = (b; i{:}s(e))$
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
<!--ID: 1713793130031-->
END%%

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@ -255,6 +255,14 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
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END%%
%%ANKI
Basic
Let $A$ and $B$ be sets. Proving the following is equivalent to showing what class is a set? $$\exists C, \forall y, (y \in C \Leftrightarrow y = \{x\} \times B \text{ for some } x \in A)$$
Back: $\{\{x\} \times B \mid x \in A\}$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1718105051820-->
END%%
## Empty Set Axiom
There exists a set having no members: $$\exists B, \forall x, x \not\in B$$

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@ -95,53 +95,130 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
<!--ID: 1717678753145-->
END%%
Given two sets $A$ and $B$, the **Cartesian product** $A \times B$ is defined as: $$A \times B = \{\langle x, y \rangle \mid x \in A \land y \in B\}$$
A **relation** $R$ is a set of ordered pairs. The **domain** of $R$ ($\mathop{\text{dom}}{R}$), the **range** of $R$ ($\mathop{\text{ran}}{R}$), and the **field** of $R$ ($\mathop{\text{fld}}{R}$) is defined as:
* $x \in \mathop{\text{dom}}{R} \Leftrightarrow \exists y, \langle x, y \rangle \in R$
* $x \in \mathop{\text{ran}}{R} \Leftrightarrow \exists t, \langle t, x \rangle \in R$
* $\mathop{\text{fld}}{R} = \mathop{\text{dom}}{R} \cup \mathop{\text{ran}}{R}$
%%ANKI
Basic
How is the Cartesian product of $A$ and $B$ denoted?
Back: $A \times B$
What is a relation?
Back: A set of ordered pairs.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1717679397781-->
<!--ID: 1718107987764-->
END%%
%%ANKI
Basic
Using ordered pairs, how is $A \times B$ defined?
Back: $\{\langle x, y \rangle \mid x \in A \land y \in B\}$
Are relations or sets the more general concept?
Back: Sets.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1717679397797-->
<!--ID: 1718107987776-->
END%%
%%ANKI
Basic
Who is attributed the representation of points in a plane?
Back: René Descartes.
How is the ordering relation $<$ on $\{2, 3, 5\}$ defined?
Back: As set $\{\langle 2, 3\rangle, \langle 2, 5 \rangle, \langle 3, 5 \rangle\}$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1717679397825-->
<!--ID: 1718107987783-->
END%%
%%ANKI
Basic
Why is the Cartesian product named the way it is?
Back: It is named after René Descartes.
Reference: “Cartesian Product,” in _Wikipedia_, April 17, 2024, [https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305).
<!--ID: 1717679397836-->
How is the ordering relation $<$ on $\{2, 3, 5\}$ visualized?
Back:
![[relation-ordering-example.png]]
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1718107987794-->
END%%
%%ANKI
Basic
Suppose $x, y \in A$. What set is $\langle x, y \rangle$ in?
Back: $\mathscr{P}\mathscr{P}A$
A relation is a set of ordered pairs with what additional restriction?
Back: N/A.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1717679397848-->
<!--ID: 1718107987803-->
END%%
%%ANKI
Cloze
{$x \in A$} iff {$\{x\} \subseteq A$} iff {$\{x\} \in \mathscr{P}A$}.
For relation $R$, {$xRy$} is alternative notation for {$\langle x, y \rangle \in R$}.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
%%ANKI
Basic
How is ordering relation $<$ on set $\mathbb{R}$ defined using set-builder notation?
Back: As $\{\langle x, y\rangle \in \mathbb{R} \times \mathbb{R} \mid x \text{ is less than } y\}$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
%%ANKI
Basic
How is $x < y$ rewritten to emphasize that $<$ is a relation?
Back: $\langle x, y \rangle \in \;<$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1718107987831-->
END%%
%%ANKI
Basic
How is the identity relation on $\omega$ defined using set-builder notation?
Back: $\{\langle n, n \rangle \mid n \in \omega\}$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
%%ANKI
Basic
How is the domain of relation $R$ denoted?
Back: $\mathop{\text{dom}}{R}$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1718107987850-->
END%%
%%ANKI
Basic
How is the domain of relation $R$ defined?
Back: $x \in \mathop{\text{dom}}{R} \Leftrightarrow \exists y, \langle x, y \rangle \in R$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
%%ANKI
Basic
How is the range of relation $R$ denoted?
Back: $\mathop{\text{ran}}{R}$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1718107987872-->
END%%
%%ANKI
Basic
How is the range of relation $R$ defined?
Back: $x \in \mathop{\text{ran}}{R} \Leftrightarrow \exists t, \langle t, x \rangle \in R$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1718107987880-->
END%%
%%ANKI
Basic
How is the field of relation $R$ denoted?
Back: $\mathop{\text{fld}}{R}$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1718107987887-->
END%%
%%ANKI
Basic
How is the field of relation $R$ defined?
Back: $\mathop{\text{fld}}{R} = \mathop{\text{dom}}{R} \cup \mathop{\text{ran}}{R}$
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
<!--ID: 1718107987897-->
END%%
## Bibliography